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Rapid Eccentricity Oscillations and the Mergers of Compact Objects In Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 29 June 2018 (MN LATEX style file v2.2) Rapid Eccentricity Oscillations and the Mergers of Compact Objects in Hierarchical Triples Joe M. Antognini1, Benjamin J. Shappee1, Todd A. Thompson1,2, Pau Amaro-Seoane3 1 Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA 2 Center of Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA 3 Max Planck Institut f¨ur Gravitationsphysik (Albert-Einstein Institut), D-14476 Potsdam, Germany E-mail: [email protected] 29 June 2018 ABSTRACT Kozai-Lidov (KL) oscillations can accelerate compact object mergers via gravita- tional wave (GW) radiation by driving the inner binaries of hierarchical triples to high eccentricities. We perform direct three-body integrations of high mass ratio compact object triple systems using Fewbody including post-Newtonian terms. We find that the inner binary undergoes rapid eccentricity oscillations (REOs) on the timescale of the outer orbital period which drive it to higher eccentricities than secular theory would otherwise predict, resulting in substantially reduced merger times. For a uniform distribution of tertiary eccentricity (e2), ∼ 40% of systems merge within ∼ 1 − 2 ec- centric KL timescales whereas secular theory predicts that only ∼20% of such systems merge that rapidly. This discrepancy becomes especially pronounced at low e2, with secular theory overpredicting the merger time by many orders of magnitude. We show that a non-negligible fraction of systems have eccentricity > 0.8 when they merge, in contrast to predictions from secular theory. Our results are applicable to high mass ratio triple systems containing black holes or neutron stars. In objects in which tidal effects are important, such as white dwarfs, stars, and planets, REOs can reduce the tidal circularization timescale by an order of magnitude and bring the components of the inner binary into closer orbits than would be possible in the secular approximation. 1 INTRODUCTION Wu & Murray 2003; Fabrycky & Tremaine 2007; Wu et al. 2007), and as a formation channel for blue stragglers Hierarchical triple systems are common (Raghavan et al. (Perets & Fabrycky 2009). 2010) and exhibit dynamics that are qualitatively differ- Kozai-Lidov cycles can drive the inner binary in some ent from binary systems (Poincar´e1892). For example, if hierarchical triple systems to merger via gravitational wave the tertiary is highly inclined with respect to the inner bi- emission (Blaes et al. 2002; Miller & Hamilton 2002). If the nary, it induces slow oscillations of the orbital parameters inner binary consists of two white dwarfs, Thompson (2011) of the inner binary. In particular, the eccentricity of the showed that these mergers occur rapidly enough to poten- inner binary oscillates between a minimum and maximum 2 tially explain the Type Ia supernova rate. Katz & Dong value (emax = 1 5/3 cos i in the limit of a test particle arXiv:1308.5682v2 [astro-ph.HE] 18 Feb 2014 − (2012) demonstrated that in a non-negligible fraction of sys- secondary when the three-body Hamiltonian is expanded p tems, perturbations to the secular Kozai-Lidov oscillations to quadrupole order) over the timescale (e.g., Holman et al. can drive the inner binary to collide head-on, rather than 1997; Blaes et al. 2002) coalescing due to gravitational radiation after tidal capture −1/2 3/2 (see also Prodan et al. 2013). Hamers et al. (2013) provide 5 m1 + m2 a1 tKL 1.3 10 yr 6 −2 a more detailed discussion of the rates of such collisions by ∼ × 2 10 M⊙ 10 pc × 3 accounting for the evolution of the inner binary on the main m1 + m2 a2/a1 2 3/2 sequence. 1 e2 . (1) × 2m2 10 − In addition to WD-WD mergers, Kozai-Lidov oscil- These oscillations are known as Kozai-Lidov oscillations lations have also been studied as a mechanism to drive (Kozai 1962; Lidov 1962). other compact objects to rapid merger. Neutron star- Kozai-Lidov oscillations have found application in a neutron star and neutron star-black hole mergers have wide variety of astrophysical systems including the orbits been proposed as engines of short gamma-ray bursts of asteroids in the Solar System (Kozai 1962), the orbits (Paczynski 1986; Ruffert et al. 1995, 1996; Ruffert & Janka of artificial satellites around planets in the Solar System 1999; Janka et al. 1999), and such mergers may also be expe- (Lidov 1962), as a formation channel for hot Jupiters (e.g., dited by Kozai-Lidov oscillations (Thompson 2011). Several c 0000 RAS 2 Antognini et al. authors have studied mergers of stellar-mass black holes, the merger time distribution and dynamics of compact ob- particularly in globular clusters, to determine if they can ject binaries. In systems with tertiaries in low eccentricity efficiently grow to intermediate-mass black holes (IMBHs; orbits we find that the double-orbit-averaged secular ap- e.g., Miller & Hamilton 2002; Wen 2003; G¨ultekin et al. proximation fails by predicting merger times many orders 2004; Aarseth 2012). Hierarchical triples of supermassive of magnitude longer than those of the direct three-body in- black holes (SMBHs) may also merge quickly as a result of tegration. Kozai-Lidov oscillations (Blaes et al. 2002; Hoffman & Loeb This paper is structured as follows. In 2 we describe § 2007; Amaro-Seoane et al. 2010). These effects can also pro- our numerical methods and characterize the accuracy of our duce interesting gravitational wave signatures from stellar integration (see also the Appendix). In 3 we describe the § mass binaries in orbit around one or more SMBHs (e.g., breakdown of the secular approximation in calculating the Antonini & Perets 2012; Nate Bode & Wegg 2013). eccentricity of the inner binary. In 4 we demonstrate one § Given its general nature, the physics behind the Kozai- regime in which this breakdown of the secular approxima- Lidov mechanism has come under broader study in the past tion leads to catastrophic failure, namely in predicting the several years. Until recently, almost all work exploring it merger times of compact objects. We conclude and discuss has employed the secular approximation, which assumes a number of applications in 5. § that any changes to the orbital parameters of the system As this manuscript was being completed, are slow compared to the orbital period of the outer bi- Antonini et al. (2013) presented similar results on the nary. The Hamiltonian is expanded in powers of the ratio of breakdown of the secular approximation, the delay time the semi-major axis of the inner binary to the semi-major distribution, and the eccentricity distribution of compact axis of the outer binary (a1/a2), typically to quadrupole or- object binaries at merger. 2 der, (a1/a2) . Krymolowski & Mazeh (1999) and Ford et al. (2000, 2004) derived the equations of motion to octupole 3 order, (a1/a2) (see Naoz et al. 2013a). Lithwick & Naoz (2011) and Katz et al. (2011) explored the implications of 2 NUMERICAL METHODS & SETUP these equations and showed that the octupole-order terms can lead to substantially larger eccentricities of the inner We numerically evolve triple systems with the open source Fewbody Fewbody binary. This so-called eccentric Kozai mechanism (EKM) suite (Fregeau et al. 2004). is designed has dramatically expanded the parameter space in which to compute the dynamics of hierarchical systems of small mergers and other interesting dynamics can occur (e.g. numbers of objects (N < 10) either in scattering experi- Naoz et al. 2012; Shappee & Thompson 2013). ments or in bound systems.∼ The underlying integrator for It is becoming increasingly evident, however, that the the Fewbody suite is the GNU Scientific Library ordinary secular approximation can fail in certain circumstances. differential equations library (Gough 2009). By default Few- Antonini & Perets (2012) found that in extreme-mass-ratio body uses eighth-order Runge-Kutta Prince-Dormand in- systems, eccentricities change rapidly compared to the pe- tegration with adaptive time steps. It is straightforward to riod of the tertiary if the tertiary is in an eccentric orbit modify Fewbody to use any of the other roughly half-dozen 2 (this behavior can also be seen in Antonini et al. 2010). integration algorithms supported by GSL. In our experi- Nate Bode & Wegg (2013) found that in a more general set ence the choice of integration algorithm does not affect the of systems, the eccentricity of the inner binary varies on the results since the adaptive steps force the size of the error to timescale of the orbit of the tertiary. Recently, Katz & Dong be within the same target value regardless of the algorithm (2012) found that these rapid variations can lead to col- used. All results in this paper were obtained using the de- lisions of WD-WD binaries if the tertiary is at very high fault eighth-order Runge-Kutta Prince-Dormand algorithm. inclination.1 Finally, Seto (2013) examined the impact of To incorporate relativistic effects, we have included post- these rapid fluctuations on gravitational wave astronomy. Newtonian (PN) terms up to order 3.5 in the integration. In this paper we revisit earlier calculations of the Details of energy conservation, gravitational radiation, and merger times of compact objects by Blaes et al. (2002) and a comparison to secular calculations are provided in Ap- Hoffman & Loeb (2007). We extend these works by directly pendix A. These additions to Fewbody and a direct ap- integrating the equations of motion of the three-body sys- plication to the formation of gravitational-wave sources for tem and including post-Newtonian (PN) force terms up to ground-based detectors will be presented in more detail in order 3.5 to account for GR effects. We show that motion of Amaro-Seoane (in prep.).
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